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The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newto- nian Casson fluid bounded by two parallel non-conducting porous plates has been studied with heat transfer considering the Hall effect. The fluid is acted upon by a uniform and exponential decaying pressure gradient. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate is suddenly set into mo- tion and simultaneously suddenly isothermally heated to a temperature other than the lower plate temperature. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of unsteady pressure gradient, the Hall term, the parameter describing the non-Newtonian behavior on both the velocities and temperature distributions have been stud- ied.

The study of Couette flow in a rectangular channel of an electrically conducting viscous fluid under the action of a transversely applied magnetic field has immediate applications in many devices such as magnetohydrodynamic (MHD) power generators, MHD pumps, accelerators, aerodynamics heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil and fluid droplets sprays. Channel flows of a Newtonian fluid with heat transfer were studied with or without Hall currents by many authors [1-10]. These results are important for the design of the duct wall and the cooling arrangements. The most important nonNewtonian fluid possessing a yield value is the Casson fluid, which has significant applications in polymer processing industries and biomechanics. Casson fluid is a shear thinning liquid which has an infinite viscosity at a zero rate of shear. Casson’s constitutive equation represents a nonlinear relationship between stress and rate of strain and has been found to be accurately applicable to silicon suspensions, suspensions of bentonite in water and lithographic varnishes used for printing inks [11-13]. Many authors [14-20] studied the flow or/and heat transfer of a non-Newtonian fluids in different geometries. The effect of time dependent pressure gradient on unsteady dusty fluid was studied by Rukmangadachari [

Attia [

The geometry of the problem is shown in _{o} while the lower plate is stationary. The upper plate is simultaneously subjected to a step change in temperature from T_{1 }to T_{2}. Then, the upper and lower plates are kept at two constant temperatures T_{2} and T_{1} respectively, with T_{2} > T_{1}. The fluid is acted upon by an exponentially decaying pressure gradient ∂p/∂x in the x-direction, and a uniform suction from above and injection from below which are applied at t = 0. A uniform magnetic field B_{o} is applied in the positive y-direction and is assumed undisturbed as the induced magnetic field is neglected by assuming a very small magnetic Reynolds number. The Hall effect is taken into consideration and consequently a z-component for the velocity is expected to arise. The uniform suction implies that the y-component of the velocity v_{0} is constant. Thus, the fluid velocity vector is given by,

The fluid motion starts from rest at t = 0, and the no-slip condition at the plates in z-direction implies that the fluid velocity has no z-component at y = ± h. The initial temperature of the fluid is assumed to be equal to T_{1}. Since the plates are infinite in the x and zdirections, the physical quantities do not change in these directions.

The flow of the fluid is governed by the momentum equation

where r is the density of the fluid and m is the apparent viscosity of the model and is given by

where K_{c}^{2} is the Casson’s coefficient of viscosity and t_{o} is the yield stress. If the Hall term is retained, the current density J is given by

where s is the electric conductivity of the fluid and b is the Hall factor [

where m is the Hall parameter and m = _{o}. Thus, the two components of the momentum Equation (1) read

where is the unsteady pressure gradient The energy equation with viscous and Joule dissipations is given by

where c_{p} and k are, respectively, the specific heat capacity and the thermal conductivity of the fluid. The second and third terms on the right-hand side represent the viscous and Joule dissipations respectively. We notice that each of these terms has two components. This is because the Hall effect brings about a velocity w in the z-direction. The initial and boundary conditions of the problem are given by

at t £ 0, and at y =-h and y =h for t > 0

u = 0 at y =-h for t > 0, u = U_{o} at y =h for t > 0, (8)

T = T_{1} at t £ 0, T = T_{2} at y =h and T = T_{1} at y = -h for t > 0 (9)

It is expedient to write the above equations in the non-dimensional form. To do this, we introduce the following non-dimensional quantities

is the constant pressure gradient.

is the decaying parameter in the unsteady pressure gradient

is the Casson number (dimensionless yield stress)

is the Reynolds number,

is the suction parameter,

is the Prandtl number,

is the Eckert number,

is the Hartmann number squared

In terms of the above non-dimensional variables and parameters Eqs.(5-9) and (2) are, respectively, written as (where the hats are dropped for convenience);

where is the constant pressure gradient and a is the decaying parameter.

Equations (10,11,15) represent coupled system of nonlinear partial differential equations which are solved numerically under the initial and boundary conditions (13) using the finite difference approximations. A linearization technique is first applied to replace the nonlinear terms at a linear stage, with the corrections incorporated in subsequent iterative steps until convergence is reached. Then the Crank-Nicolson implicit method is used at two successive time levels [

where

and

The variables with bars are given initial guesses from the previous time step and an iterative scheme is used at every time to solve the linearized system of difference equations. Then the finite difference form for the energy Equation (12) can be written as

where DISP represents the Joule and viscous dissipation terms which are known from the solution of the momentum equations and can be evaluated at the mid point (i+1/2,j+1/2) of the computational cell. Computations have been made for α = 5, Pr = 1, Re = 1, Ha = 3 and Ec = 0.2. Grid-independence studies show that the computational domain 0 ¥ and –1< y <1 can be divided into intervals with step sizes Dt = 0.0001 and Dy = 0.005 for time and space respectively. The truncation error of the central difference schemes of the governing equations is . Stability and rate of convergence are func-

tions of the flow and heat parameters. Smaller step sizes do not show any significant change in the results. Convergence of the scheme is assumed when all of the unknowns u, v, w, B, θ and H for the last two approximations differ from unity by less than 10^{-6} for all values of y in –1 < y < 1 at every time step. Less than 7 approximations are required to satisfy this convergence criteria for all ranges of the parameters studied here.

Figures 3-5 show the variation of the velocity components u and w and the temperature θ at the central

plane of the channel (y = 0) with time. These figures show the results for various values of the decaying parameter a = 0, 1 and 2 and for yield stress t_{D }= 0.0, 0.05 and 0.1. In these figures S = 1 and m = 3. _{D}. It is observed also that the time at which u reaches its steady state value decreases with increasing a for a > 0 but that occurs earlier for constant pressure gradient (a = 0). Increasing t_{D} increases u for all values of a but with

small differences.

In _{D} on w depends on t and becomes more clear when the decaying parameter a = 0 but this influence is small for large a. It is observed that increasing t_{D} more decreases w for a = 0.

while it is not greatly affected by changing t_{D}. The figure shows also that the time at which θ reaches its steady state value decreases with increasing a while it is not greatly affected by changing t_{D}.

Figures 6-8 show the profiles of the velocity components u and w and the temperature θ, respectively, for various values of time a and for t = 0.2, 1, and 2. The figures are evaluated for m =3, t_{D} = 0.05 and S = 1. It is clear from Figures 6 and 7 that the effect of decaying

parameter a on u and w depends on t and y.

the velocity u.

monotonically. Increasing a decreases θ for all y and t. It is observed also that the velocity component u reaches the steady state faster than w which, in turn, reaches the steady state faster than θ. This is expected as u is the source of w, while both u and w act as sources for the temperature.

Figures 9-11 show the variation of the velocity components u and w and the temperature θ at the central

plane of the channel (y = 0) with time for various values of the Hall parameter m and for t_{D }= 0.0, 0.05, and 0.1. In these figures S = 0. _{D} as the effective conductivity (s /(1+m^{2})) decreases with increasing m which reduces the magnetic damping force on u. It is observed also from the figure that the time at which u reaches its steady state value increases with increasing m. Increasing

t_{D} increases u for all m and its effect on u becomes more pronounced for higher values of m. In ^{2}) ) which decreases with increasing m (m > 1 ). This accounts for

the crossing of the curves of w with t for all values of t_{D}. _{D} on w depends on t and becomes more clear when m is large. It is observed that, increasing t_{D} decreases w and increasing m increases w.

but this is reversed at large times. This is due to the fact that, for small times, u and w are small and an increase in m increases u but decreases w. Then, the Joule dissipation which is also proportional to (1/1+m^{2}) decreases. For large times, increasing m increases both u and w and, in turn, increases the Joule and viscous dissipations. This accounts for the crossing of the curves of θ with time for

all values of t_{D}. It is also observed that increasing t_{D} decreases the temperature θ for all values of m. This is because increasing t_{D} decreases both u and w and their gradients which decreases the Joule and viscous dissipations. The figure shows also that the time at which θ reaches its steady state value increases with increasing m while it is not greatly affected by changing t_{D}.

Figures (12-14) present the profiles of the velocity components u and w and the temperature θ for variousvalues of time t and for t_{D }= 0.0, 0.05 and 0.1. The fig-

ures are evaluated for m = 3 and S = 1. It is clear from Figures 12 and 13 that the effect of yield stress t_{D} on u and w depends on t and y. _{D} decreases u for small y, but this is reversed for large y. As time develops, increasing t_{D} increases u for all y. _{D} increases w for all values of y, but increasing t_{D} more decreases w for large y.

For large t, increasing t_{D} decreases w for small t and all values of y. This can be attributed to the fact that increasing t_{D} will delay the attainment of maxima of u and w. It is also observed, from Figures 12 and 13, that the velocity components u and w do not reach their steady state monotonically. _{D} increases θ for all y and t as a result of increasing the dissipations. It is observed also that the velocity component u reaches the steady state faster than w which, in turn, reaches the steady state faster than θ. This is expected as u is the source of w, while both u and w act as sources for the temperature.

A finite difference method is used to solve the transient Couette flow and heat transfer of a Casson non-Newtonian fluid under the influence of unsteady pressure gradient and uniform magnetic field. In the present work, we study Hall effect. The effects of the decaying parameter a, Casson yield stress t_{D}, and the Hall parameter m on the velocity and temperature distributions are studied. The decaying parameter a affects the main velocity components u and w and the temperature θ. The Hall term affects the main velocity component u in the x-direction and gives rise to another velocity component w in the z-direction. The results show that the influence of the parameters a and t_{D} on u and w depend on time and the Hall parameter m. It is also found that the effect of m on w and θ depends on time for all values of t_{D} which accounts for a crossover in the w-t and θ-t graphs for various values of m. The effect of m on the magnitude of θ depends on t_{D} and becomes more pronounced in case of small t_{D}. It is also found that the effect of a on the magnitude of θ depends on t_{D} and becomes more pronounced in case of smallt_{D.}